Ann. Geophys., 31, 1437–1446, 2013www.ann-geophys.net/31/1437/2013/doi:10.5194/angeo-31-1437-2013© Author(s) 2013. CC Attribution 3.0 License.

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DiscussionsDispersion relation of electromagnetic ion cyclotron waves usingCluster observations

I. P. Pakhotin1, S. N. Walker1, Y. Y. Shprits2,3,4, and M. A. Balikhin 1

1Department of Automatic Control and Systems Engineering, University of Sheffield, Sheffield, South Yorkshire, UK2Department of Earth and Space Sciences, UCLA, Los Angeles, CA, USA3Department of Atmospheric and Planetary Sciences, MIT, Cambridge, MA, USA4Skolkovo Institute of Science and Technology, Moscow, Russian Federation

Correspondence to:I. P. Pakhotin (mea07ip@sheffield.ac.uk)

Received: 14 January 2013 – Revised: 5 July 2013 – Accepted: 18 July 2013 – Published: 26 August 2013

Abstract. Multi-point wave observations on Cluster space-craft are used to infer the dispersion relation of electromag-netic ion cyclotron (EMIC) waves. In this study we use aphase differencing method and observations from STAFFand WHISPER during a well-studied event of 30 March2002. The phase differencing method requires the knowledgeof the direction of the wave vector, which was obtained usingminimum variance analysis. Wave vector amplitudes werecalculated for a number of frequencies to infer the dispersionrelation experimentally. The obtained dispersion relation islargely consistent with the cold plasma dispersion relation.The presented method allows inferring the dispersion rela-tion experimentally. It can be also used in the future to anal-yse the hot plasma dispersion relation of waves near the localgyrofrequency that can occur under high plasma beta condi-tions.

Keywords. Magnetospheric physics (plasmasphere; plasmawaves and instabilities)

1 Introduction

The near-Earth radiation belt environment presents a numberof hazards to orbiting spacecraft (Baker, 1996). A better un-derstanding of the nature of the magnetosphere dynamics andassociated processes will in turn allow a better understandingof these dangers, allowing engineers to design spacecraft towithstand such effects. Understanding and modelling radia-tion belt dynamics helps mitigate spacecraft damage.

The Van Allen radiation belts are formed from energeticcharged particles that are trapped by the Earth’s magnetic

field. In the inner magnetosphere, there are broadly twozones of enhanced trapped particle flux. The inner and outerbelts are separated by a slot region at around 2–3RE whereenergetic particle flux levels are usually low. The existenceof the slot region is a result of the balance between an in-ward radial diffusion of electrons from the outer belt andprecipitation losses due to resonant scattering by very lowfrequency (VLF) waves, primarily plasmaspheric hiss (e.g.Lyons et al., 1972; Lyons and Thorne, 1973; Selesnick et al.,2003; Thorne et al., 2007; Shprits et al., 2008b). The fluxesof high energy particles in the inner belt are relatively sta-ble, with a predictable variation linked to the solar cycle (Liet al., 2001). However, the relativistic electron fluxes in theouter belt are highly variable, and can change over an orderof magnitude in a few hours (e.g.Thorne et al., 2007).

Radiation belt enhancements appear to be correlated withgeomagnetic storm activity, but the link between storm activ-ity and radiation belt enhancement is complicated, with rel-ativistic electron flux increases observed only in about halfof the cases studied (Reeves et al., 2003). One-fifth of allstorms analysed produced flux decreases, with the rest leav-ing flux levels largely unchanged. Clearly, several competingprocesses exist during times of enhanced geomagnetic activ-ity that produce both particle acceleration and loss. In theirreview, Friedel et al.(2002) identified several mechanismsfor the build-up of energetic (MeV) electrons in the mag-netosphere. Most sources listed in the review fall broadlyinto two categories: those which rely on inward radial dif-fusion (Kellogg, 1959; Tverskoy, 1964, 1965; Falthammar,1965; Elkington et al., 2003), and those with a local ac-celeration mechanism. Radial diffusion can be enhanced by

Published by Copernicus Publications on behalf of the European Geosciences Union.

1438 I. P. Pakhotin et al.: Dispersion relation of EMIC waves using Cluster observations

magnetospheric pulsations (e.g.Thorne, 2010) which aregenerated by large-scale disturbances such as the Kelvin–Helmholtz instability on the flanks of the magnetosphere(Shprits et al., 2008a, and references therein), buffeting ofthe magnetosphere on the day side, or by convective injec-tions of protons on the night side. Evidence for local acceler-ation as a source of relativistic electrons was found by stormmodelling (Summers et al., 2002). Later,Chen et al.(2006)conducted an analysis of the radial profiles of phase spacedensity (PSD) distributions of energetic electrons using com-bined satellite measurements and found evidence for both theexternal (radial transport) and internal (wave–particle inter-actions) processes acting simultaneously. The study estab-lished a peak in the radial PSD profile inside the outer radia-tion belt, consistent with a local acceleration source. Furtherstudies (Shprits et al., 2007) demonstrated the importance ofthe local stochastic acceleration of particles by their inter-action with VLF waves, in particular chorus waves, duringmagnetic storm recovery.

Wave–particle interactions can transfer energy from onepart of the energetic particle distribution to another, result-ing in local particle acceleration. Change of the pitch angleresulting from wave–particle interactions can lead to the pre-cipitation via the loss cone. Thus plasma waves can causeboth flux enhancement and depletion.

Electromagnetic ion cyclotron (EMIC) waves are excitedin the Pc 1–2 range (period 0.2–10 s) just below each plasmaion species gyrofrequency usually in the region of the mag-netic equator (e.g.Fraser et al., 1996). It is commonly ac-cepted that these waves are excited by anisotropic ring cur-rent ions injected into the inner magnetosphere during sub-storms (Summers and Thorne, 2003; Li et al., 2007) or byaforementioned changes in the plasma distribution functionproduced by lower frequency waves (e.g.Fraser et al., 1992;Thorne, 2010). EMIC waves propagate and grow aroundthe dense plasmasphere region near the plasmapause (Fraserand Nguyen, 2001), cyclotron-resonate with radiation beltions and Landau-resonate with outer radiation belt electrons(Horne and Thorne, 1990, 1994) scattering both species intothe loss cone. Thus they are believed responsible for both ra-diation belt depletions and ring current ion losses (e.g.Fraseret al., 2010). They provide possibly the dominant mechanismfor scattering electrons into the loss cone in the inner mag-netosphere (Shprits et al., 2008b). While there are many re-ports of EMIC waves observations based on the spectra ofobserved turbulence, this study is devoted to the determina-tion of the dispersion relation of EMIC waves directly frommulti-point Cluster observations.

Since the initial analytical work (Stix, 1962; Smith andBrice, 1964) identifying the dispersion surfaces for ion cy-clotron waves, there have been numerous theoretical workson ion cyclotron dispersion relations. It is now known thatsignificant numbers of heavy ions such as He+ and O+ ex-ist in the inner magnetosphere, and so the dispersion re-lations for multiple-ion plasmas in the presence of heavy

ions have been theoretically calculated (Smith and Brice,1964; Young et al., 1981; Swanson, 1989; Horne and Thorne,1990). However, there has been relatively little experimen-tal work to confirm the theory and ray tracing simulationsdirectly. Broughton et al.(2008) used the wave telescopemethod to analyse ultra-low frequency (ULF) waves in theplasma sheet boundary layer. This paper uses the cospec-tral/phase differencing method ofk vector determination(Balikhin and Gedalin, 1993; Dudok de Wit et al., 1995; Ba-likhin et al., 1997a) to calculate the frequency–wavenumberrelation directly and then compare it to existing EMIC wavedispersion theory. This method assumes that in the plasmarest frame (PF) a wave field can be represented asB(r, t)=

6ωB(ω)expj (kr −ωt)+C, whereC is the complex con-jugate term. The PF frequencyω and the wave vectork arerelated by the dispersion relationω = ω(k). If two satellitesare separated by a vectorR, the phase shift1ψ between thetwo measured time series at the observed frequencyω1 is es-timated as

1ψ(ω1)= (k(ω1)R)= |k(ω1)||R|cos(θkR). (1)

Thus the phase shift is a function of the scalar product ofk

andR. The result is a projection ofk on R; to calculate thefull k vector, knowledge of its direction is required. In thecase of magnetic field measurements, the minimum variancemethod (Sonnerup and Cahill, 1967) can be applied to iden-tify the direction of the wave vector. This method has the lim-itation that it can only be used for circularly/elliptically po-larised waves. Since EMIC waves are often left-hand circularpolarised at generation (e.g.Young et al., 1981), minimumvariance can be exploited in this study. Finally, knowledgeof the wave vector for any observed frequencyω1 “enablesthe determination of the wave dispersion using the Dopplerrelation:ω1 = ω+kV 0(V 0 is the plasma bulk velocity)” (Ba-likhin et al., 1997a). The phase differencing techniques havebeen exploited and enabled identification of wave modes invarious regions of geospace such as foreshock, shock frontand the magnetosheath (Balikhin et al., 1997b; Chishamet al., 1999; Walker et al., 2004).

2 Instrumentation

The Cluster satellites were launched in 2000 into a polar or-bit with period of 57 h, apogee outside the bow shock or deepin the magnetotail (depending on the season) and perigee inthe inner magnetosphere. The mission consists of 4 space-craft flying in a tetrahedral formation, allowing simultaneousmulti-satellite measurements to resolve spatial and temporaleffects. The Cluster fluxgate magnetometer (FGM) instru-ment (Balogh et al., 2001) provides measurements of mag-netic fluctuations for the events studied in the present pa-per at a sampling frequency of 22 Hz, making it suitable forEMIC wave analysis. The Cluster ion spectrometry (CIS)instrument (Reme et al., 2001) provides measurements of

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I. P. Pakhotin et al.: Dispersion relation of EMIC waves using Cluster observations 1439

Fig. 1. Plasmapause crossing on 30 March 2002 as viewed from Cluster 1. Graphs, from top to bottom, show(a) energetic ion densityrecorded by CIS,(b) ion velocity from CIS,(c) magnetic field strength from FGM,(c) magnetic field waveform spectra from STAFF and(d) electric field waveform spectra from WHISPER. The inbound crossing is at 07:30, and the outbound crossing is at 08:45 UTC. EMICwaves can be seen on STAFF, the fourth panel from 08:00 until 08:30 UTC.

ion moments (mass, density, velocity, temperature paralleland perpendicular to the local magnetic field) for several ionspecies. The STAFF search coil magnetometers (Cornilleau-Wehrlin et al., 2003) provide waveform data of the magneticfield sampled at 25 Hz. WHISPER (Decreau et al., 1997)measures the electric field up to 80 kHz in active and passivemodes. The STAFF and WHISPER instruments form partof the Wave Experiment Consortium (WEC) complex thatis controlled by Digital Wave Processing experiment (DWP)(Woolliscroft et al., 1997).

3 Dispersion determination methodology

Time series data or Morlet wavelet spectral plots are analysedfor all four satellites for the duration of an event. Individualwave packets are identified by the shape of their envelope.By visual inspection it can in some cases be determined withcertainty that two wave packets observed on different satel-lites are in fact the same wave packet propagating from onesatellite to another. In these cases a histogram of the phasedifference as a function of frequency is calculated using theMorlet wavelet transform. Each wave packet shows up asa discrete frequency with an associated wave vector mag-nitude, which is simply1ψ/|R|. There is a 2π ambiguity

which is due to a periodicity in phase difference and shouldstrictly speaking extend to±∞. The correct1ψ branch canbe identified from either time series data or dynamical spec-trum plots, by observing which satellite first sees the wavepower rise over the noise level. For each wave packet, thespectrum maximum represents a point on the dispersion re-lation graph. The procedure is repeated for all identifiablewave packets to obtain a set of points, which are then plottedonto a frequency–wavenumber graph. Doppler effects shouldbe taken into account, but in the case of the event studied inthis paper, the plasma velocity in the plasmasphere is aboutan order of magnitude lower than the wave velocity. Dopplereffects are therefore insignificant.

Each wavenumber obtained in this manner is not thewavenumber in the strict sense. Rather, it is the magnitude ofa vector projection of thek vector onto the inter-satellite sep-aration vectorR. To determine the fullk vector, knowledgeof the wave direction is required. The wave vector directionis obtained using minimum variance on the left-hand circu-larly polarised (LHCP) wave packets (Sonnerup and Cahill,1967). Minimum variance analysis makes use of the Maxwellequation that states that magnetic fields have zero divergence.Thus the direction where the magnetic field varies least is as-sumed to be the direction of the wave vector. The true wave

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1440 I. P. Pakhotin et al.: Dispersion relation of EMIC waves using Cluster observations

vector is then calculated as|k| = (1ψ/|R|)/θkR, whereθkRis the angle between the wave vector and the separation vec-tor.

4 Cluster observation of EMIC waves on 30 March 2002

The emissions under study took place between 08:00:50 and08:16:30 UT. Figure 1 displays an overview plot of (a) theenergetic ion density, (b) ion velocity, (c) the magnitude ofthe magnetic field, (d) dynamic spectrum of the magneticfield (STAFF), and (e) the dynamic spectrum of the electricfield obtained by WHISPER, as observed by Cluster 1 space-craft during time period 06:00–12:00 UT on 30 March 2002.During this time interval the Cluster spacecraft entered theplasmasphere around 07:30 UT. This can be seen implicitlyby the sudden drop in plasma velocity and a rise in ener-getic ion density. The magnetic field dips as the spacecraftapproaches the equator, then rises again after the equato-rial crossing as the spacecraft moves towards higher latitudeswith stronger magnetic field strength. The inbound plasma-pause crossing can also be clearly identified by the rise inthe electron plasma frequency seen by WHISPER until it isout of range of the instrument. The outbound crossing can beseen at 08:45 UT as indicated by a corresponding rapid dropin electron plasma frequency. Multiple EMIC waves wereencountered while the spacecraft were inside the plasmas-phere. The EMIC waves can be seen on the STAFF dynamicspectrum while the spacecraft were inside the plasmapause asenhancements in the wave power in the 2–3 Hz range. Dur-ing the time period of interest, the average separation vec-tor between C2 and C4 was [−4, 26,−83] km, between C1and C2 [−12, 47, 183] km and between C1 and C4 [−16,73, 100] km in the geocentric solar ecliptic (GSE) referenceframe.

The four Cluster spacecraft crossed the magnetic equa-tor at about 08:05 UT on 30 March 2002 atL≈ 4.3 (Pickettet al., 2010). The emissions were observed in the dusk/pre-midnight time sector (22:15 MLT) where there is a rather lowprobability of EMIC instabilities at theseL values (e.g.An-derson et al., 1992; Meredith et al., 2003). The event is well-documented in previous research (Pickett et al., 2010; Omuraet al., 2010; Shoji et al., 2011; Grison et al., 2013). A polari-sation analysis (Pickett et al., 2010) showed the occurrence ofmostly left-hand circularly polarised (LHCP) waves of dis-tinct frequencies separated by spectral stop bands – a typi-cal feature of EMIC waves observed in the post-noon sector(e.g.Young et al., 1981; Horne and Thorne, 1994). Figure 2shows a Morlet wavelet spectrogram from FGM on Cluster 1for the time period 08:00–08:20, with wave energy first seenat around 1.5 Hz rising to about 3 Hz, and about 12 min intothe event, energy below≈ 1 Hz (the local helium ion gyrofre-quency�He+) with reduced energy at the gyrofrequency it-self.

The time series of magnetic field data of all 4 satelliteswere analysed, to identify the time intervals during whichthe same wave packets were observed simultaneously by atleast two spacecraft so the phase differencing methodologycould be used to calculate the magnitude of thek vector. Fig-ure 3 shows one example of magnetic waveforms for a seriesof wave packets. It is clear from the amplitude envelopes ofsome of these wave packets (as highlighted by the black box)that the same waves are observed on satellites C1 and C4.The phase shift for each wave packet is visually noticeableand corresponds to the propagation delay from one satelliteto the other. In total, 21 such wave packets were analysed.Since the satellites were separated mainly along thez axisin the GSE reference frame, the mostly field-aligned equato-rial wave packets were usually observed to propagate with asmall angle (< 15◦) between the wave vector and separationvectorθkR. This meant that there was a clearly identifiablephase difference1ψ (Fig. 4). Equation (1) was applied tocalculate the projection of thek vector onR.

The wave vector was projected onto the magnetic fieldvector to calculate the angleθBk. The results indicated mostlyfield-aligned propagation, as was expected of the equatorialEMIC waves. Figure 5 shows a hodogram from the samewave packet as highlighted in Fig. 3. For easier viewing ofthe hodogram, only a segment of the original wave packetwas used. The hodogram is obtained by transforming themagnetic field time series from GSE coordinates onto thevariance frame, and allows the determination of wave po-larisation. The circular polarisation is clearly visible in thebottom panel where the direction of minimum variance is di-rected out of the page. All wave packets used in this studywere found to be left-hand circularly polarised. The methodemployed byFowler et al.(1967) was used to confirm po-larisation further. After about 08:15 UTC, linearly polarisedwave packets appeared as the constellation moved away fromthe magnetic equator, as predicted by theory and previousobservations (e.g.Young et al., 1981; Horne and Thorne,1994). Minimum variance on the linear wave packets wasnot performed due to the limitations of the (Sonnerup andCahill, 1967) method mentioned earlier. In theory it wouldhave been possible to use the minimum variance free methoddescribed byBalikhin et al.(2000) to calculate the fullk vec-tor. However, this method requires the same wave packet tobe observed on 4 satellites. Unfortunately, none of the wavepackets identified were observed on all 4 spacecraft simulta-neously.

For the 14 circularly polarised wave packets (listed in Ta-ble 1), the relation between frequency and wave vector issummarised in Fig. 6. Thex axis, labelled K, correspondsto the wave vector magnitude|k| multiplied by the proton in-ertial length c

ωpi, while they axis, labelled X, is the wave fre-

quency normalised by the proton gyrofrequency�H+ . Con-ventions are taken fromYoung et al.(1981). Following themethodology ofOmura et al.(2010), the plasma composition

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I. P. Pakhotin et al.: Dispersion relation of EMIC waves using Cluster observations 1441

Fig. 2. Frequency–time wavelet spectrogram of ULF waves observed on 30 March 2002, on Cluster 1 (By component of magnetic field)from FGM. The colour bar is in arbitrary units representative of relative power spectral density. An area of reduced wave energy can beseen around 1 Hz at the local�He+ corresponding to a stop band. The white lines, from top to bottom, represent�H+ , �He+ and�O+

respectively.

Fig. 3.Time series data of fluxgate magnetometer (FGM) instrument on board Cluster 1 (blue), Cluster 2 (red), Cluster 3 (black) and Cluster 4(magenta) showing wave packets 49 s after 08:00 UTC.By axis shifted+5 nT for Cluster 3 and−5 nT for Cluster 4 for better viewing. Theblack rectangle is around the 2 simultaneous measurements which were compared.

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1442 I. P. Pakhotin et al.: Dispersion relation of EMIC waves using Cluster observations

Fig. 4. The joint frequency–1ψ spectrum of theBy component ofthe magnetic field, for the same wave packets as pictured in Fig. 3.The colour bar is in arbitrary units representative of the relativepower spectral density as a function of frequency and phase differ-ence. The periodicity of the spectrum corresponds to an ambiguityof 2πn in the phase difference.

was assumed to be as follows:ne = 178/cc,nH+ = 144/cc,nHe+ = 17/cc,nO+ = 17/cc. A very similar composition, butinvolving energetic protons, was assumed byShoji et al.(2011) in a simulation involving the same time period as thepresent study. The magnetic field magnitude averaged overthe time period 08:00:50–08:16:30 and all 4 spacecraft was253 nT.

The theoretical cold plasma dispersion is taken from theequation for theL mode (Stix, 1962), which corresponds tofield-aligned, left-hand polarised propagation:

N2= L= 1−

∑i

ω2pi

ω2

(ω

ω− εi�i

). (2)

In Eq. (2),N2 is the refractive index (N2= k2c2/ω2), ωpi

the plasma frequency of the charged particle speciesi, �ithe cyclotron frequency of the charged particle speciesi, andsummation is over the electrons and each positive ion speciespresent in the plasma.εi is the charge state and is−1 forelectrons and+1 for each ion species. In the calculation ofthe cold plasma dispersion relation, the plasma was assumedto contain only three species of ions (H+, He+ and O+). Theresulting theoretical dispersion surface branches can be seenin Fig. 6 as solid curves. From Eq. (2) is can be seen thatin a multi-ion plasma a resonance will exist at each positiveion species gyrofrequency. At these resonance points, the therefractive index and wave vector will approach infinity. Theplotted analytical dispersion curves reflect this.

Table 1. Cluster satellite pairs, starting and ending times for eachwave packet. Often segments of the wave packets were used inthe analysis, rather than whole wave packets. This was done wherenoise or other waves contaminated sections of the wave packet timeseries. In practice, only a few oscillations are necessary to determinevector direction from minimum variance analysis.

Sat 1 Sat 2 Start time End time

C1 C4 08:00:58 08:01:05C1 C4 08:01:12 08:01:17C1 C2 08:08:41 08:08:43C1 C2 08:08:45 08:08:48C2 C4 08:12:17 08:12:19C2 C4 08:12:26 08:12:29C2 C4 08:12:30 08:12:33C2 C4 08:12:33 08:12:36C2 C4 08:12:40 08:12:44C2 C4 08:13:08 08:13:12C2 C4 08:13:12 08:13:15C2 C4 08:15:43 08:15:51C2 C4 08:15:57 08:16:05C2 C4 08:16:16 08:16:25

5 Discussion and conclusions

The points in Fig. 6 in general follow the theoretical disper-sion curves for cold plasma with the assumed ion composi-tion. Two branches are evident, above and below the heliumion gyrofrequency. As the dispersion surface approaches theH+ resonance, the normalised|k| approaches infinity asω

�H+

approaches unity. In practice collecting data near this pointmay be difficult: as the wavenumber approaches infinity, thephase velocity approaches zero and so the same wave packetmay be seen on two satellites with a significant delay, mak-ing matching and phase difference calculations inaccurate.The results in Fig. 6 correspond well with the cold plasmaapproximation where left-handed waves cannot propagatein the stop band; this is also confirmed by the FGM data(Fig. 2).

Potential limitations of the methodology are summarisedas follows. The cold plasma dispersion relation as out-lined in Stix (1962) assumes completely field-aligned wavepropagation. The phase differencing method employingwavelet transforms generates an area of high signal in fre-quency/phase space from which the maximum is taken as thedata point value. It is possible that the true phase/frequencyvalues do not correspond to that maximum. Plasma compo-sition calculations are only estimations based on the cut-offfrequency and cyclotron frequencies. The method used byOmura et al.(2010) assumes that helium and oxygen ionsexist in equal proportions in the plasma. This is required tomake the algebra a system of two equations with two un-knowns. While CIS data show that energetic oxygen and he-lium ions do exist in similar proportions, total oxygen and

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I. P. Pakhotin et al.: Dispersion relation of EMIC waves using Cluster observations 1443

Fig. 5. Hodogram representation of a fragment of the same wave packet as in Fig. 3 seen from the Cluster 4 fluxgate magnetometer data.Bmax, Bint andBmin signify maximum, intermediate and minimum variance eigenvectors respectively. The bottom panel shows the waveprojected along the axis of minimum variance (coming out of the page). The red cross marks the beginning of the time series. Left-handcircular polarisation is clearly visible.

Fig. 6. Frequency vs. wavenumber relation plot of wave packetsobserved during the event. The bold line marks the helium ion gy-rofrequency. The crosses signify individual wave packet observa-tions overplotted with error bars in frequency and wavenumber. Thesolid curves are different branches of the cold plasma dispersion re-lation for the aforementioned plasma composition.

helium ion numbers are not necessarily equal. Assuming thecold plasma approximation at all ignores the role playedby warm ions. While this is a valid assumption for a low-beta plasma regime such as encountered in this event, warmplasma effects can significantly modify dispersion surfaces

in general (Silin et al., 2011). Extraneous magnetic pulsa-tions over the same time period as the wave can contam-inate the minimum variance analysis and introduce uncer-tainty in wave vector direction, which in turn may result inan erroneousθkR and an uncertainty about the wavenumber.The magnetic field strength varies slightly for the duration ofthe event, while the Morlet wavelet spectral plot makes pre-cise identification of the cut-off frequency problematic. Theabove errors in turn cause uncertainty in calculating plasmacomposition. The plasma electron resonant frequency is outof range of the WHISPER instrument, meaning that the totalelectron density has to be computed indirectly (Pickett et al.,2010, and references therein). Despite these limitations theresults fit the theoretical data fairly well. Estimated inaccura-cies are represented by error bars in Fig. 6.

The assumption that EMIC emissions cannot propagate inthe stop band only holds for low plasma beta regimes. Workby Silin et al. (2011) suggests that, under high-beta condi-tions, EMIC waves can be excited anywhere in the stop band.The cold plasma approximation then breaks down, and thedispersion should be calculated using the full dispersion rela-tion. Such an estimation requires a detailed knowledge of notonly the densities of each species but also knowledge of thefull distribution function of each of the species. The methoddescribed in this study can be used to derive the dispersionfunction empirically for any conditions from observations,providing the wave vector can be determined. As such it can

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1444 I. P. Pakhotin et al.: Dispersion relation of EMIC waves using Cluster observations

be a useful tool to validate theoretical dispersion functionsunder unusual plasma regimes.

For the first time multiple satellite measurements havebeen used in conjunction with the phase differencing methodto compare directly the experimental and the theoretical dis-persion relation of EMIC waves just below the H+ gyrofre-quency. The evolution of normalised wave frequency withwave vector is similar to the analytical results obtained un-der the linear dispersion theory forLmode waves. Some dis-crepancies are probably due to the difficulty in ascertainingexact ion ratios.

Acknowledgements.We acknowledge the ESA Cluster ActiveArchive and the STAFF, FGM, WHISPER and CIS teams forsupplying the Cluster data. We acknowledge discussions withRichard Horne, Richard Boynton and Andrew Dimmock. The re-search leading to these results was funded by STFC. Y. Y. Shpritswould like to acknowledge UC Lab Fee Grant and NASA grantsNNX10AK99G; NNX10AK99G

Topical Editor I. A. Daglis thanks two anonymous referees fortheir help in evaluating this paper.

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